### Abstract

We consider the multi-point boundary value problem - f{symbol}_{p} (u^{'})^{'} = ? f{symbol}_{p} (u), on (- 1, 1),u (± 1) = underover(?, i = 1, m^{±}) a_{i}^{±} u (?_{i}^{±}), where p > 1, f{symbol}_{p} (s) {colon equals} | s |^{p - 1} sgn s for s ? R, ? ? R, m^{±} = 1 are integers, ?_{i}^{±} ? (- 1, 1), 1 = i = m^{±}, and the coefficients a_{i}^{±} satisfy underover(?, i = 1, m^{±}) | a_{i}^{±} | < 1 . A number ? ? R is said to be an eigenvalue of the above problem if there exists a non-trivial solution u. The spectrum is the set of eigenvalues. In this paper we obtain some basic spectral and degree-theoretic properties of this eigenvalue problem. These results have numerous applications to more general problems. As an example, a Rabinowitz-type, global bifurcation theorem is briefly described. © 2010 Elsevier Ltd. All rights reserved.

Original language | English |
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Pages (from-to) | 4244-4253 |

Number of pages | 10 |

Journal | Nonlinear Analysis: Theory, Methods and Applications |

Volume | 72 |

Issue number | 11 |

DOIs | |

Publication status | Published - 1 Jun 2010 |

### Keywords

- Eigenvalues
- Multi point boundary value problem
- p-Laplacian
- Topological degree theory